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Math Help - Isomorphism in normal subgroups

  1. #1
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    Isomorphism in normal subgroups

    Let G = Z4 + U(4), H = <(2,3)> and K = <(2,1)>. Show that G/H is not isomorphic to G/K.
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  2. #2
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    Quote Originally Posted by o&apartyrock View Post

    Let G = \mathbb{Z}_4 \oplus U(4), \ H = <(2,3)> and K = <(2,1)>. Show that G/H is not isomorphic to G/K.
    Hint: note that both G/H and G/K have 4 elements. show that every non-identity element of G/K has order 2 and find an element of G/H which has order 4.

    this is not a part of solution but the above also proves that G/K \cong \mathbb{Z}_2 \oplus \mathbb{Z}_2 and G/H \cong \mathbb{Z}_4.
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  3. #3
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    How is it that G/H and G/k both have 4 element? My calculations show them having 8 elements each, since Z4 + U(4) has 8 elements. Am I incorrect?
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  4. #4
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    Quote Originally Posted by o&apartyrock View Post

    How is it that G/H and G/k both have 4 element? My calculations show them having 8 elements each, since Z4 + U(4) has 8 elements. Am I incorrect?
    H and K have 2 elements. so |G/K|=|G|/|K|=8/2 = 4. same for G/H.
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  5. #5
    Super Member Gamma's Avatar
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    LaGrange

    Do you have LaGrange's Thorem yet?
    |G/H|=\frac{|G|}{|H|}

    As you noted |G|=|\mathbb{Z}_4 \oplus U_4|=8
    Direct computation shows |H|=|<2,3>|=|\{(2,3) , (0,1) \}|= 2
    Similarly,
    |K|=|<2,1>|=|\{(2,1) , (0,1) \}|= 2

    So you see by the above,
    |G/H|=\frac{|G|}{|H|}=\frac{8}{2}=4
    |G/H|=\frac{|G|}{|K|}=\frac{8}{2}=4

    Follow NCA's hints to see why H \not \cong K
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  6. #6
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    Ah I see. Simple mistake on my part.
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